2020 AIME II Problem 15

Let \triangle A B C be an acute scalene triangle with circumcircle \omega. The tangents to \omega at B and C intersect at T. Let X and Y be the projections of T onto lines A B and A C, respectively. Suppose B T=C T=16, B C=22, and T X^{2}+T Y^{2}+X Y^{2}=1143. Find X Y^{2}.